Formula:KLS:01.05:06

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\HyperpFq ⁢ 54 ⁢ @ ⁢ @ ⁢ 1 + a / 2 , a , b , c , d ⁢ a / 2 , 1 + a - b , 1 + a - c , 1 + a - d ⁢ 1 = Γ ⁡ ( 1 + a - b ) ⁢ Γ ⁡ ( 1 + a - c ) ⁢ Γ ⁡ ( 1 + a - d ) ⁢ Γ ⁡ ( 1 + a - b - c - d ) Γ ⁡ ( 1 + a ) ⁢ Γ ⁡ ( 1 + a - b - c ) ⁢ Γ ⁡ ( 1 + a - b - d ) ⁢ Γ ⁡ ( 1 + a - c - d ) \HyperpFq 54 @ @ 1 𝑎 2 𝑎 𝑏 𝑐 𝑑 𝑎 2 1 𝑎 𝑏 1 𝑎 𝑐 1 𝑎 𝑑 1 Euler-Gamma 1 𝑎 𝑏 Euler-Gamma 1 𝑎 𝑐 Euler-Gamma 1 𝑎 𝑑 Euler-Gamma 1 𝑎 𝑏 𝑐 𝑑 Euler-Gamma 1 𝑎 Euler-Gamma 1 𝑎 𝑏 𝑐 Euler-Gamma 1 𝑎 𝑏 𝑑 Euler-Gamma 1 𝑎 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle{}\HyperpFq{5}{4}@@{1+a/2,a,b,c,d}{a% /2,1+a-b,1+a-c,1+a-d}{1}{}=\frac{\Gamma\left(1+a-b\right)\Gamma\left(1+a-c% \right)\Gamma\left(1+a-d\right)\Gamma\left(1+a-b-c-d\right)}{\Gamma\left(1+a% \right)\Gamma\left(1+a-b-c\right)\Gamma\left(1+a-b-d\right)\Gamma\left(1+a-c-d% \right)}}}}

Proof

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Symbols List

F q p subscript subscript 𝐹 𝑞 𝑝 {\displaystyle{\displaystyle{\displaystyle{{}_{p}F_{q}}}}}  : generalized hypergeometric function : http://dlmf.nist.gov/16.2#E1
Γ Γ {\displaystyle{\displaystyle{\displaystyle\Gamma}}}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1

Bibliography

Equation in Section 1.5 of KLS.

URL links

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